Math 240: Transition to Advanced Math Deductive reasoning: logic is used to draw conclusions based on statements accepted as true. Thus conclusions are proved to be true, provided assumptions are true. If results are incorrect, then assumptions need to be modified. In this course we focus on logic and proof, as opposed to computational methods such as algebra and calculus.

Ch 1.1: Propositions & Connectives Proposition: A sentence that is either true or false. Examples: Which is a proposition? Today is Monday = x = 5 What is mathematics? Newton wore green socks occasionally. This sentence is false.

Connectives Definitions: Given two propositions P and Q, The conjunction of P and Q is denoted by P /\ Q, and represents the proposition “P and Q.” P /\ Q is true exactly when P and Q are true. The disjunction of P and Q is denoted by P \/ Q, and represents the proposition “P or Q.” P \/ Q is true exactly when at least one of P or Q is true. The negation of P is denoted by ~P, and represents the proposition “not P.” ~P is true exactly when P is false.

Examples Suppose P = “Chickens ride the bus” and Q = “2 > 1” Find the truth value of the following propositions P /\ Q P \/ Q ~P ~Q

Propositional Forms The sentence “Chickens ride the bus or 2 >1” is a proposition, while the symbolic representation “P \/ Q” is a propositional form. (Compare counting with algebra) A propositional form is an expression involving finitely many logical symbols and letters. The truth value of a propositional form can be found using a truth table.

Truth Tables A truth table must list all possible combinations of truth values of components of propositional form. Example: Give the truth table for P /\ Q. Example: Give the truth table for P \/ Q. PQP /\ Q TTT FTF TFF FFF PQP \/ Q TTT FTT TFT FFF

Truth Tables Example: Give the truth table for ~P. Example: Give the truth table for (P /\ Q) \/ ~Q Example: Give the truth table for P \/ (Q /\ R) Example: Find the truth value of (P \/ S) /\ (P \/ T), given that P is true while S and T are false.

Equivalence Definition: Two propositions P and Q are equivalent iff they have the same truth table. Example: P is equivalent to P \/ (P /\ Q) Example: P is equivalent to ~(~P)

Denial Definition: A denial of a proposition S is any proposition equivalent to ~S. Example: Suppose P = “4 is an odd number” ~P = “It is not the case that 4 is an odd number” Useful denials: 4 is not odd 4 is even The remainder when dividing 4 by 2 is 0 Example: Cleopatra was an excellent Math 240 student. Explain.

Denial Does ~(P \/ Q) = (~P) \/ (~Q)? Does ~(P /\ Q) = (~P) /\ (~Q)? You will find out in the homework!

Order of Operations Use delimiters such as ( ), { }, [ ], in the usual way. Next First priority: ~ Second: /\ Third: \/ Example ~P \/ Q = (~P) \/ Q P \/ Q /\ R = P \/ (Q /\ R) Left to right priority: P \/ Q \/ R = (P \/ Q) \/ R P \/ Q /\ R \/ ~R = ( P \/ [Q /\ R]) \/ (~R) Parentheses are good, but can get unwieldy.

Tautologies Definition: A tautology is a propositional form that is true for every assignment of truth values of components. Example: P \/ ~P is a tautology P~PP \/ ~P TFT FTT

Contradiction Definition: A contradiction is a propositional form that is false for every assignment of truth values of components. Example: P /\ ~P is a contradiction P~PP /\ ~P TFF FTF

Homework Read Ch 1.1 Do 7(1,2a-e,i,3a,b,d-g,j,k,4a-c,h,5a-d,6a,b,7,11a,b)